Non-Asymptotic Analysis of Classical Spectrum Estimators for $L$-mixing Time-series Data with Estimated Means

Abstract

Spectral estimation is an important tool in time series analysis, with applications including economics, astronomy, and climatology. The asymptotic theory for non-parametric estimation is well-known but the development of non-asymptotic theory is still ongoing. Our recent work obtained the first non-asymptotic error bounds on the Bartlett and Welch methods with restrictive assumptions. In this work, we derive non-asymptotic error bounds for both Bartlett and Welch estimators for $L$-mixing time-series data with unknown means, and the results cover the special case with known zero means. The class of $L$-mixing processes contains common models in time series analysis, including autoregressive processes and measurements of geometrically ergodic Markov chains. Our new error bounds are of $O(\frac{1}{\sqrt{k}})$, where $k$ is the number of data segments used in the algorithm. Such bounds are the tightest among the existing work on non-asymptotic analysis of classical spectrum estimators with or without zero-mean assumptions.

11 pages, 2 figures, under review